Probabilistic proof of the linear discrepancy bound

Establish Theorem 1, namely the bound F(P) \geq n/3 for every noncollinear n-point set P in the plane with switches corresponding to all connecting lines, using the probabilistic method.

Background

The paper proves that, for every noncollinear n-point set P in the plane and every initial assignment of \pm 1 weights, a sequence of switches along connecting lines can produce signed discrepancy at least n/3. This contrasts with the classical Gale–Berlekamp game, whose asymptotic bounds are obtained using probabilistic arguments.

The authors explicitly state that they do not know how to derive their geometric theorem by the probabilistic method. This is a concrete unresolved methodological problem concerning an alternative proof of the paper’s main result.